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You can use the Phased Array System Toolbox™ software to simulate radar systems that transmit, propagate, reflect, and receive polarized electromagnetic fields. By including this capability, the toolbox can realistically model the interaction of radar waves with targets and the environment.

It is a basic property of plane waves in free-space that the
directions of the electric and magnetic field vectors are orthogonal
to their direction of propagation. The direction of propagation of
an electromagnetic wave is determined by the *Poynting* vector

$$S=E\times H$$

$$\begin{array}{ll}E\hfill & =-Zs\times H\hfill \\ H\hfill & =\frac{1}{Z}s\times E\hfill \end{array}$$

After manipulating the two equations, you can see that the electric and magnetic fields are orthogonal to the direction of propagation

$$E\xb7s=H\xb7s=0.$$

$$E={E}_{x}{\widehat{e}}_{x}+{E}_{y}{\widehat{e}}_{y}$$

The unit vectors together with the unit vector in direction of propagation

$$\{{\widehat{e}}_{x},{\widehat{e}}_{y},s\}\text{.}$$

For a radar system, the electric and magnetic field are actually
spherical waves, rather than plane waves. However, in practice, these
fields are usually measured in the far field region or radiation zone
of the radar source and are approximately plane waves. In the far
field, the waves are called *quasi-plane* waves.
A point lies in the *far field* if its distance, *R*,
from the source satisfies *R ≫D ^{2}/λ* where

Polarization applies to purely sinusoidal signals. The most general expression for a sinusoidal plane-wave has the form

$$E={E}_{x0}\mathrm{cos}\left(\omega t-k\xb7x+{\varphi}_{x}\right){\widehat{e}}_{x}+{E}_{y0}\mathrm{cos}\left(\omega t-k\xb7x+{\varphi}_{y}\right){\widehat{e}}_{y}={E}_{x}{\widehat{e}}_{x}+{E}_{y}{\widehat{e}}_{y}$$

The quantities *E _{x0}* and

You can usually suppress the spatial dependence of the field and write the electric field vector as

$$E={E}_{x0}\mathrm{cos}\left(\omega t+{\varphi}_{x}\right){\widehat{e}}_{x}+{E}_{y0}\mathrm{cos}\left(\omega t+{\varphi}_{y}\right){\widehat{e}}_{y}={E}_{x}{\widehat{e}}_{x}+{E}_{y}{\widehat{e}}_{y}$$

The preceding equation for a polarized plane wave shows that
the tip of the two-dimensional electric field vector moves along a
path which lies in a plane orthogonal to field’s direction
of propagation. The shape of the path depends upon the magnitudes
and phases of the components. For example, if *ϕ _{x}* =

$${E}_{y}=\frac{{E}_{y0}}{{E}_{x0}}{E}_{x}$$

This equation represents a straight line through the origin
with positive slope. Conversely, suppose *ϕ _{x} =
ϕ_{y} + π*. Then, the tip
of the electric field vector follows a straight line through the origin
with negative slope

$${E}_{y}=-\frac{{E}_{y0}}{{E}_{x0}}{E}_{x}$$

A different case occurs when the amplitudes are the same, *E _{x}* =

$$\begin{array}{ll}{E}_{x}\hfill & ={E}_{0}\mathrm{cos}(\omega t+\varphi )\hfill \\ {E}_{y}\hfill & ={E}_{0}\mathrm{cos}(\omega t+\varphi \pm \pi /2)=\mp {E}_{0}\mathrm{sin}(\omega t+\varphi )\hfill \end{array}$$

$${E}_{x}^{2}+{E}_{y}^{2}={E}_{0}^{2}$$

While this equation gives the path the vector takes, it does
not tell you in what direction the electric field vector travels around
the circle. Does it rotate clockwise or counterclockwise? The rotation
direction depends upon the sign of *π/2* in
the phase. You can see this dependency by examining the motion of
the tip of the vector field. Assume the common phase angle, *ϕ
= 0*. This assumption is permissible because the common phase
only determines starting position of the vector and does not change
the shape of its path. First, look at the *+π/2* case
for a wave travelling along the *s*-direction (out
of the page). At *t=0*, the vector points along the *x*-axis.
One quarter period later, the vector points along the negative *y-*axis.
After another quarter period, it points along the negative *x-*axis.

MATLAB^{®} uses the IEEE convention to assign the
names *right-handed* or *left-handed* polarization
to the direction of rotation of the electric vector, rather than *clockwise* or *counterclockwise*.
When using this convention, left or right handedness is determined
by pointing your left or right thumb along the direction of propagation
of the wave. Then, align the curve of your fingers to the direction
of rotation of the field at a given point in space. If the rotation
follows the curve of your left hand, then the wave is left-handed
polarized. If the rotation follows the curve of your right hand, then
the wave is right-handed polarized. In the preceding scenario, the
field is left-handed circularly polarized (LHCP). The phase difference *–π/2* corresponds
to right-handed circularly polarized wave (RHCP). The following figure
provides a three-dimensional view of what a LHCP electromagnetic wave
looks like as it moves in the *s*-direction.

When the terms *clockwise* or *counterclockwise* are
used they depend upon how you look at the wave. If you look along the direction of propagation,
then the clockwise direction corresponds to right-handed polarization and counterclockwise
corresponds to left-handed polarization. If you look toward where the wave is coming from, then
clockwise corresponds to left-handed polarization and counterclockwise corresponds to
right-handed polarization.

**Left-Handed Circular Polarization**

The figure below shows the appearance of linear and circularly polarized fields as they move
towards you along the *s*-direction.

**Linear and Circular Polarization**

Besides the linear and circular states of polarization, a third
type of polarization is *elliptic polarization*.
Elliptic polarization includes linear and circular polarization as
special cases.

As with linear or circular polarization, you can remove the time dependence to obtain the locus of points that the tip of the electric field vector travels

$${\left(\frac{{E}_{x}}{{E}_{x0}}\right)}^{2}+{\left(\frac{{E}_{y}}{{E}_{y0}}\right)}^{2}-2\left(\frac{{E}_{x}}{{E}_{x0}}\right)\left(\frac{{E}_{y}}{{E}_{y0}}\right)\mathrm{cos}\varphi ={\mathrm{sin}}^{2}\varphi $$

The size and shape of a two-dimensional ellipse can be defined
by three parameters. These parameters are the lengths of its two axes,
the semi-major axis, *a*, and semi-minor axis, *b*,
and a tilt angle, *τ*. The following figure
illustrates the three parameters of a tilted ellipse. You can derive
them from the two electric field amplitudes and phase difference.

**Polarization Ellipse**

Polarization can best be understood in terms of complex signals. The complex representation of a polarized wave has the form

$$E={E}_{x0}{e}^{i{\varphi}_{x}}{e}^{i\omega t}{\widehat{e}}_{x}+{E}_{y0}{e}^{i{\varphi}_{y}}{e}^{i\omega t}{\widehat{e}}_{y}=\left({E}_{x0}{e}^{i{\varphi}_{x}}{\widehat{e}}_{x}+{E}_{y0}{e}^{i{\varphi}_{y}}{\widehat{e}}_{y}\right){e}^{i\omega t}$$

$$\rho =\frac{{E}_{y0}}{{E}_{x0}}{e}^{i\left({\varphi}_{y}-{\varphi}_{x}\right)}=\frac{{E}_{y0}}{{E}_{x0}}{e}^{i\varphi}$$

It is useful to introduce the *polarization vector*.
For the complex polarized electric field above, the polarization vector, ** P**, is obtained by normalizing the electric
field

$$P=\frac{{E}_{x0}}{{E}_{m}}{\widehat{e}}_{x}+\frac{{E}_{y0}}{{E}_{m}}{e}^{i\left({\varphi}_{y}-{\varphi}_{x}\right)}{\widehat{e}}_{y}=\frac{{E}_{x0}}{{E}_{m}}{\widehat{e}}_{x}+\frac{{E}_{y0}}{{E}_{m}}{e}^{i\varphi}{\widehat{e}}_{y}$$

The overall size of the polarization ellipse is not important
because that can vary as the wave travels through space, especially
through geometric attenuation. What is important is the shape of the
ellipse. Thus, the significant ellipse parameters are the ratio of
its axis dimensions, *a/b*, called the *axial
ratio*, and the *tilt angle*, *τ*.
Both of these quantities can be determined from the ratio of the component
amplitudes and the phase difference, or, equivalently, from the polarization
ratio. Another quantity, equivalent to the axial ratio, is the *ellipticity
angle*, *ε*.

In the Phased Array
System Toolbox software, you can use the `polratio`

function to convert the complex
amplitudes `fv=[Ey;Ex]`

to the polarization ratio.

p = polratio(fv)

The tilt angle is defined as the positive (counterclockwise)
rotation angle from the *x*-axis to the semi-major
axis of the ellipse. Because of the symmetry properties of the ellipse,
the tilt angle, *τ*, needs only be defined
in the range *–π/2 ≤ τ ≤
π/2*. You can find the tilt angle by determining the
rotated coordinate system in which the semi-major and semi-minor axes
align with the rotated coordinate axes. Then, the ellipse equation
has no cross-terms. The solution takes the form

$$\mathrm{tan}2\tau =\frac{2{E}_{x0}{E}_{y0}}{{E}_{x0}^{2}-{E}_{y0}^{2}}\mathrm{cos}\varphi $$

After solving for the tilt angle, you can determine the semi-major
and semi-minor axis lengths. Conceptually, you rotate the ellipse
clockwise by the tilt angle and measure the lengths of the intersections
of the ellipse with the *x*- and *y*-axes.
The point of intersection with the larger value is the semi-major
axis, *a*, and the one with the smaller value is
the semi-minor axis, *b*.

The *axial ratio* is defined as *AR
= a/b* and, by construction, is always greater than or equal
to one. The *ellipticity angle* is defined by

$$\mathrm{tan}\epsilon =\mp \frac{b}{a}$$

If you first introduced the *auxilliary angle*, *α*,
by

$$\mathrm{tan}\alpha =\frac{{E}_{y0}}{{E}_{x0}}$$

$$\mathrm{sin}2\epsilon =\mathrm{sin}2\alpha \mathrm{sin}\varphi $$

For elliptic polarization, just as with circular polarization,
you need another parameter to completely describe the ellipse. This
parameter must provide the rotation sense or the direction that the
tip of the electric (or magnetic vector) moves in time. The rate of
change of the angle that the field vector makes with the *x*-axis
is proportion to *–sin φ* where *φ* is
the phase difference. If *sin φ* is positive,
the rate of change is negative, indicating that the field has left-handed
polarization. If *sin φ* is negative, the rate
of change is positive or right-handed polarization.

The function `polellip`

lets
you find the values of the parameters of the polarization ellipse
from either the field component vector `fv=[Ey;Ex]`

or
the polarization ratio, `p`

.

fv=[Ey;Ex]; [tau,epsilon,ar,rs] = polellip(fv); p = polratio(fv); [tau,epsilon,ar,rs] = polellip(p);

`tau`

, `epsilon`

, `ar`

and `rs`

represent
the tilt angle, ellipticity angle, axial ratio and rotation sense,
respectively. Both syntaxes give the same result.This table summaries several different common polarization states and the values of the amplitudes, phases, and polarization ratio that produce them:

Polarization | Amplitudes | Phases | Polarization Ratio |
---|---|---|---|

Linear positive slope | Any non-negative real values for E._{x},
E_{y} | φ_{y} = φ_{x} | Any non-negative real number |

Linear negative slope | Any non-negative real values for E_{x},
E_{y} | φ_{y} = φ_{x}+
π | Any negative real number |

Right-Handed Circular | E_{x}=E_{y} | φ_{y}= φ_{x}–
π/2 | –i |

Left-Handed Circular | E_{x}=E_{y} | φ_{y}= φ_{x} +
π/2 | i |

Right-Handed Elliptical | Any non-negative real values for E_{x},
E_{y} | sin (φ_{y}– φ_{x})
< 0 | sin(arg ρ) < 0 |

Left-Handed Elliptical | Any non-negative real values for E_{x},
E_{y} | sin (φ_{y}– φ_{x})
>0 | sin(arg ρ) > 0 |

As shown earlier, you can express a polarized electric field
as a linear combination of basis vectors along the *x* and *y* directions.
For example, the complex electric field vectors for the right-handed
circularly polarized (RHCP) wave and the left-handed circularly polarized
(LHCP) wave, take the form:

$$E=\mathrm{Re}[{E}_{0}({e}_{x}\mp i{e}_{y}){e}^{i(\omega t+\varphi )}]$$

$$\begin{array}{l}{e}_{r}=\frac{1}{\sqrt{2}}({e}_{x}-i{e}_{y})\\ {e}_{l}=\frac{1}{\sqrt{2}}({e}_{x}+i{e}_{y})\end{array}$$

$$\begin{array}{ll}{e}_{x}\hfill & =\frac{1}{\sqrt{2}}({e}_{r}+{e}_{l})\hfill \\ {e}_{y}\hfill & =\frac{1}{\sqrt{2}i}({e}_{r}-{e}_{l})\hfill \end{array}$$

$$E={E}_{l}{e}_{l}+{E}_{r}{e}_{r}$$

The polarized field is orthogonal to the wave’s direction
of propagation. Thus, the field can be completely specified by the
two complex components of the electric field vector in the plane of
polarization. The formulation of a polarized wave in terms of two-component
vectors is called the *Jones vector* formulation.
The Jones vector formulation can be expressed in either a linear basis
or a circular basis or any basis. This table shows the representation
of common polarizations in a linear basis and circular basis.

Common Polarizations | Jones Vector in Linear Basis | Jones Vector in Circular Basis |
---|---|---|

Vertical | `[0;1]` | `1/sqrt(2)*[-1;1]` |

Horizontal | `[1;0]` | `1/sqrt(2)*[1;1]` |

45° Linear | `1/sqrt(2)*[1;1]` | `1/sqrt(2)*[1-1i;1+1i]` |

135° Linear | `1/sqrt(2)*[1;-1]` | `1/sqrt(2)*[1+1i;1-1i]` |

Right Circular | `1/sqrt(2)*[1;-1i]` | `[0;1]` |

Left Circular | `1/sqrt(2)*[1;1i]` | `[1;0]` |

The polarization ellipse is an instantaneous representation of a polarized wave. However, its parameters, the tilt angle and the ellipticity angle, are often not directly measurable, particularly at very high frequencies such as light frequencies. However, you can determine the polarization from measurable intensities of the polarized field.

The measurable intensities are the Stokes parameters, *S _{0}*,

$$\begin{array}{ll}{S}_{0}\hfill & ={E}_{x0}^{2}+{E}_{y0}^{2}\hfill \\ {S}_{1}\hfill & ={E}_{x0}^{2}-{E}_{y0}^{2}\hfill \\ {S}_{2}\hfill & =2{E}_{x0}{E}_{y0}\mathrm{cos}\varphi \hfill \\ {S}_{3}\hfill & =2{E}_{x0}{E}_{y0}\mathrm{sin}\varphi \hfill \end{array}$$

For completely polarized fields, you can show by time averaging the polarization ellipse equation that

$${S}_{0}^{2}={S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}$$

For partially polarized fields, in contrast, the Stokes parameters satisfy the inequality

$${S}_{0}^{2}<{S}_{1}^{2}+{S}_{3}^{2}+{S}_{3}^{2}$$

The Stokes parameters are related to the tilt and ellipticity
angles, *τ* and *ε*

$$\begin{array}{ll}{S}_{1}\hfill & ={S}_{0}\mathrm{cos}2\tau \mathrm{cos}2\epsilon \hfill \\ {S}_{2}\hfill & ={S}_{0}\mathrm{sin}2\tau \mathrm{cos}2\epsilon \hfill \\ {S}_{3}\hfill & ={S}_{0}\mathrm{sin}2\epsilon \hfill \end{array}$$

$$\begin{array}{ll}\mathrm{tan}2\tau \hfill & =\frac{{S}_{2}}{{S}_{1}}\hfill \\ \mathrm{sin}2\epsilon \hfill & =\frac{{S}_{3}}{{S}_{0}}\hfill \end{array}$$

The two-dimensional Poincaré sphere can help you visualize
the state of a polarized wave. Any point on or in the sphere represents
a state of polarization determined by the four Stokes parameters, *S _{0},
S_{1}, S_{2}*, and

As an example, solve for the Stokes parameters of a RHCP field, `fv=[1,-i]`

,
using the `stokes`

function.

S = stokes(fv)

S = 2 0 0 -2

Antennas couple propagating electromagnetic radiation to electrical currents in wires, electromagnetic fields in waveguides or aperture fields. This coupling is a phenomenon common to both transmitting and receiving antennas. For some transmitting antennas, source currents in a wire produce electromagnetic waves that carrying power in all directions. Sometimes an antenna provides a means for a guided electromagnetic wave on a transmission line to transition to free-space waves such as a waveguide feeding a dish antennas. For receiving antennas, electromagnetic fields can induce currents in wires to generate signals to be then amplified and passed on to a detector.

For transmitting antennas, the shape of the antenna is chosen
to enhance the power projected into a given direction. For receiving
antennas, you choose the shape of the antenna to enhance the power
received from a particular direction. Often, many transmitting antennas
or receiving antennas are formed into an *array*.
Arrays increase the transmitted power for a transmitting system or
the sensitivity for a receiving system. They improve directivity over
a single antenna.

An antenna can be assigned a polarization. The polarization of a transmitting antenna is the polarization of its radiated wave in the far field. The polarization of a receiving antenna is actually the polarization of a plane wave, from a given direction, resulting in maximum power at the antenna terminals. By the reciprocity theorem, all transmitting antennas can serve as receiving antennas and vice versa.

Each antenna or array has an associated local Cartesian coordinate
system *(x,y,z)* as shown in the following figure.
See Global and Local Coordinate Systems for more information.
The local coordinate system can also be represented by a spherical
coordinate system using azimuth, elevation and range coordinates, *az,
el, r*, or alternately written, *(φ,θ,r)*,
as shown. At each point in the far field, you can create a set of
unit spherical basis vectors, $$\{{\widehat{e}}_{H},{\widehat{e}}_{V},\widehat{r}\}$$. The basis vectors
are aligned with the *(φ,θ,r)* directions,
respectively. In the far field, the electric field is orthogonal to
the unit vector $$\widehat{r}$$. The components of a polarized
field with respect to this basis, *(E _{H},E_{V})*,
are called the horizontal and vertical components of the polarized
field. In radar, it is common to use

$$E=F(\varphi ,\theta )\frac{{e}^{ikr}}{r}=\left({F}_{H}(\varphi ,\theta ){\widehat{e}}_{H}+{F}_{V}(\varphi ,\theta ){\widehat{e}}_{V}\right)\frac{{e}^{ikr}}{r}$$

The simplest polarized antenna is the dipole antenna which consist
of a split length of wire coupled at the middle to a coaxial cable.
The simplest dipole, from a mathematical perspective, is the *Hertzian* dipole,
in which the length of wire is much shorter than a wavelength. A diagram
of the short dipole antenna of length *L* appears
in the next figure. This antenna is fed by a coaxial feed which splits
into two equal length wires of length *L/2*. The
current, *I*, moves along the *z*-axis
and is assumed to be the same at all points in the wire.

The electric field in the far field has the form

$$\begin{array}{l}{E}_{r}=0\\ {E}_{H}=0\\ {E}_{V}=-\frac{i{Z}_{0}IL}{2\lambda}\mathrm{cos}\text{el}\text{\hspace{0.22em}}\frac{{e}^{-ikr}}{r}\end{array}$$

The next example computes the vertical and horizontal polarization components of the field. The vertical component is a function of elevation angle and is axially symmetric. The horizontal component vanishes everywhere.

The toolbox lets you model a short dipole antenna using the `phased.ShortDipoleAntennaElement`

System
object™.

**Short-Dipole Polarization Components**

Compute the vertical and horizontal polarization components of the field created by a short-dipole antenna pointed along the *z*-direction. Plot the components as a function of elevation angle from 0° to 360°.

**Note:** This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent `step`

syntax. For example, replace `myObject(x)`

with `step(myObject,x)`

.

Create the `phased.ShortDipoleAntennaElement`

System object™.

antenna = phased.ShortDipoleAntennaElement(... 'FrequencyRange',[1,2]*1e9,'AxisDirection','Z');

Compute the antenna response. Because the elevation angle argument to `antenna`

is restricted to ±90°, compute the responses for 0° azimuth and then for 180° azimuth. Combine the two responses in the plot. The operating frequency of the antenna is 1.5 GHz.

el = [-90:90]; az = zeros(size(el)); fc = 1.5e9; resp = antenna(fc,[az;el]); az = 180.0*ones(size(el)); resp1 = antenna(fc,[az;el]);

Overlay the responses in the same figure.

figure(1) subplot(121) polar(el*pi/180.0,abs(resp.V.'),'b') hold on polar((el+180)*pi/180.0,abs(resp1.V.'),'b') str = sprintf('%s\n%s','Vertical Polarization','vs Elevation Angle'); title(str) hold off subplot(122) polar(el*pi/180.0,abs(resp.H.'),'b') hold on polar((el+180)*pi/180.0,abs(resp1.H.'),'b') str = sprintf('%s\n%s','Horizontal Polarization','vs Elevation Angle'); title(str) hold off

The plot shows that the horizontal component vanishes, as expected.

You can use a cross-dipole antenna to generate circularly-polarized
radiation. The crossed-dipole antenna consists of two identical but
orthogonal short-dipole antennas that are phased 90° apart. A
diagram of the crossed dipole antenna appears in the following figure.
The electric field created by a crossed-dipole antenna constructed
from a *y*-directed short dipole and a *z*-directed
short dipole has the form

$$\begin{array}{l}{E}_{r}=0\\ {E}_{H}=-\frac{i{Z}_{0}IL}{2\lambda}\mathrm{cos}\text{az}\text{\hspace{0.22em}}\frac{{e}^{-ikr}}{r}\\ {E}_{V}=\frac{i{Z}_{0}IL}{2\lambda}(\mathrm{sin}\text{el}\mathrm{sin}\text{az+}i\mathrm{cos}\text{el})\frac{{e}^{-ikr}}{r}\end{array}$$

The polarization ratio *E _{V}/E_{H}*,
when evaluated along the

The toolbox lets you model a crossed-dipole antenna using the `phased.CrossedDipoleAntennaElement`

System
object.

**LHCP and RHCP Polarization Components**

This example plots the right-handed and left-handed circular polarization components of fields generated by a crossed-dipole antenna at 1.5 GHz. You can see how the circular polarization chnages from pure RHCP at 0° azimuth angle to pure LHCP at 180° azimuth angle, both at 0° elevation angle.

**Note:** This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent `step`

syntax. For example, replace `myObject(x)`

with `step(myObject,x)`

.

Create the `phased.CrossedDipoleAntennaElement`

System object™.

```
fc = 1.5e9;
antenna = phased.CrossedDipoleAntennaElement('FrequencyRange',[1,2]*1e9);
```

Compute the left-handed and right-handed circular polarization components from the antenna response.

az = [-180:180]; el = zeros(size(az)); resp = antenna(fc,[az;el]); cfv = pol2circpol([resp.H.';resp.V.']); clhp = cfv(1,:); crhp = cfv(2,:);

Plot both circular polarization components at 0° elevation.

polar(az*pi/180.0,abs(clhp)) hold on polar(az*pi/180.0,abs(crhp)) title('LHCP and RHCP vs Azithmuth Angle') legend('LHCP','RHCP') hold off

You can create polarized fields from arrays by using polarized
antenna elements as a value of the `Elements`

property
of an array System
object. All Phased Array
System Toolbox arrays
support polarization.

After a polarized field is created by an antenna system, the
field radiates to the far-field region. When the field propagates
into free space, the polarization properties remain unchanged until
the field interacts with a material substance which scatters the field
into many directions. In such situations, the amplitude and polarization
of the scattered wave can differ from the incident wave polarization.
The scattered wave polarization may depend upon the direction in which
the scattered wave is observed. The exact way that the polarization
changes depends upon the properties of the scattering object. The
quantity describing the response of an object to the incident field
is called the radar scattering cross-section matrix (RSCM), *S*.
You can measure the scattering matrix as follows. When a unit amplitude
horizontally polarized wave is scattered, both a horizontal and a
vertical scattered component are produced. Call these two components *S _{HH}* and

$$\left[\begin{array}{c}{E}_{H}^{(scat)}\\ {E}_{V}^{(scat)}\end{array}\right]=\sqrt{\frac{4\pi}{{\lambda}^{2}}}\left[\begin{array}{cc}{S}_{HH}& {S}_{VH}\\ {S}_{HV}& {S}_{VV}\end{array}\right]\left[\begin{array}{c}{E}_{H}^{(inc)}\\ {E}_{V}^{(inc)}\end{array}\right]=\sqrt{\frac{4\pi}{{\lambda}^{2}}}\left[S\right]\left[\begin{array}{c}{E}_{H}^{(inc)}\\ {E}_{V}^{(inc)}\end{array}\right]$$

To understand how the scattered wave depends upon the polarization of the incident wave, you need to examine all possible scattered field polarizations for each incident polarization. Because this amount of data is difficult to visualize, consider two cases:

For the

*copolarization*case, the scattered polarization has the same polarization as the incident field.For the

*cross-polarization*case, the scattered polarization has an orthogonal polarization to the incident field.

You can represent the incident polarizations in terms of the tilt angle-ellipticity angle pair $$\left(\tau ,\epsilon \right)$$. Every unit incident polarization vector can be expressed as

$$\left[\begin{array}{c}{E}_{H}^{(inc)}\\ {E}_{V}^{(inc)}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\tau & -\mathrm{sin}\tau \\ \mathrm{sin}\tau & \mathrm{cos}\tau \end{array}\right]\left[\begin{array}{c}\mathrm{cos}\epsilon \\ j\mathrm{sin}\epsilon \end{array}\right]$$

$$\left[\begin{array}{c}{E}_{H}^{(inc)\perp}\\ {E}_{V}^{(inc)\perp}\end{array}\right]=\left[\begin{array}{cc}-\mathrm{sin}\tau & -\mathrm{cos}\tau \\ \mathrm{cos}\tau & -\mathrm{sin}\tau \end{array}\right]\left[\begin{array}{c}\mathrm{cos}\epsilon \\ -j\mathrm{sin}\epsilon \end{array}\right]$$

$${P}^{(co)}={\left[\begin{array}{cc}{E}_{H}^{(inc)}& {E}_{V}^{(inc)}\end{array}\right]}^{*}S\left[\begin{array}{c}{E}_{H}^{(inc)}\\ {E}_{V}^{(inc)}\end{array}\right]$$

`[]*`

denotes
complex conjugation. To obtain the cross-polarization signature, compute$${P}^{(cross)}={\left[\begin{array}{cc}{E}_{H}^{(inc)\perp}& {E}_{V}^{(inc)\perp}\end{array}\right]}^{*}S\left[\begin{array}{c}{E}_{H}^{(inc)}\\ {E}_{V}^{(inc)}\end{array}\right]$$

`polsignature`

function. This function
returns the absolute value of the scattered power (normalized by its
maximum value). The next example shows how to plot the polarization
signatures for the RSCM matrix $$S=\left[\begin{array}{cc}2i& \frac{1}{2}\\ \frac{1}{2}& i\end{array}\right]$$

Plot the copolarization and cross-polarization signatures of the scattering matrix

Specify the scattering matrix. and specify the range of ellipticity angles and orientation (tilt) angles that define the polarization states. These angles cover all possible incident polarization states.

rscmat = [1i*2,0.5;0.5,-1i]; el = [-45:45]; tilt = [-90:90];

Plot the copolarization signatures for all incident polarizations.

`polsignature(rscmat,'c',el,tilt)`

Plot the cross-polarizations signatures for all incident polarizations.

`polsignature(rscmat,'x',el,tilt)`

An antenna that is used to receive polarized electromagnetic waves achieves its maximum output power when the antenna polarization is matched to the polarization of the incident electromagnetic field. Otherwise, there is polarization loss:

The polarization loss is computed from the projection (or dot product) of the transmitted field’s electric field vector onto the receiver polarization vector.

Loss occurs when there is a mismatch in direction of the two vectors, not in their magnitudes.

The polarization loss factor describes the fraction of incident power that has the correct polarization for reception.

Using the transmitter’s spherical basis at the receiver’s
position, you can represent the incident electric field, *(E _{iH},
E_{iV})*, by

$$E={E}_{iH}{\widehat{e}}_{H}+{E}_{iV}{\widehat{e}}_{V}={E}_{m}{P}_{i}$$

You can represent the receiver’s polarization vector, *(P _{H},
P_{V})*, in the receiver’s local
spherical basis by:

$$P={P}_{H}{{\widehat{e}}^{\prime}}_{H}+{P}_{V}{{\widehat{e}}^{\prime}}_{V}$$

The next figure shows the construction of the transmitter and receiver spherical basis vectors.

The polarization loss is defined by:

$$\rho =\frac{|{E}_{i}\cdot P{|}^{2}}{\left|{E}_{i}{|}^{2}\right|P{|}^{2}}$$

`polloss`

computes the polarization mismatch
between an incident field and a polarized antenna.To achieve maximum output power from a receiving antenna, the matched antenna polarization vector must be the complex conjugate of the incoming field’s polarization vector. As an example, if the incoming field is RHCP, with polarization vector given by $${e}_{r}=\frac{1}{\sqrt{2}}({e}_{x}-i{e}_{y})$$, the optimum receiver antenna polarization is LHCP. The introduction of the complex conjugate is needed because field polarizations are described with respect to its direction of propagation, whereas the polarization of a receive antenna is usually specified in terms of the direction of propagation towards the antenna. The complex conjugate corrects for the opposite sense of polarization when receiving.

As an example, if the transmitting antenna transmits an RHCP field, the polarization loss factors for various received antenna polarizations are

Receive Antenna Polarization | Receive Antenna Polarization Vector | Polarization Loss Factor | Polarization Loss Factor (dB) |
---|---|---|---|

Horizontal linear | e_{H} | 1/2 | 3 dB |

Vertical linear | e_{V} | 1/2 | 3 |

RHCP | $${e}_{r}=\frac{1}{\sqrt{2}}({e}_{x}-i{e}_{y})$$ | 0 | ∞ |

LHCP | $${e}_{l}=\frac{1}{\sqrt{2}}({e}_{x}+i{e}_{y})$$ | 1 | 0 |

This example models a tracking radar based on a 31-by-31 (961-element) uniform rectangular array (URA). The radar is designed to follow a moving target. At each time instant, the radar points in the known direction of the target. The basic radar requirements are the probability of detection, `pd`

, the probability of false alarm, `pfa`

, the maximum unambiguous range, `max_range`

, and the range resolution, `range_res`

, (all distance units are in meters). The `range_gate`

parameter limits the region of interest to a range smaller than the maximum range. The operating frequency is set in `fc`

. The simulation lasts for `numpulses`

pulses.

**Radar Definition**

Set up the radar operating parameters. The existing radar design meets the following specifications.

pd = 0.9; % Probability of detection pfa = 1e-6; % Probability of false alarm max_range = 1500*1000; % Maximum unambiguous range range_res = 50.0; % Range resolution rangegate = 5*1000; % Assume all objects are in this range numpulses = 200; % Number of pulses to integrate fc = 8e9; % Center frequency of pulse c = physconst('LightSpeed'); tmax = 2*rangegate/c; % Time of echo from object at rangegate

**Pulse Repetition Interval**

Set the pulse repetition interval, `PRI`

, and pulse repetition frequency, `PRF`

, based on the maximum unambiguous range.

PRI = 2*max_range/c; PRF = 1/PRI;

**Transmitted Signal**

Set up the transmitted rectangular waveform using the `phased.RectangularWaveform`

System object(TM). The waveform pulse width, `pulse_width`

, and pulse bandwidth, `pulse_bw`

, are determined by the range resolution you select,|range_res|. Specify the sampling rate, `fs`

, to be twice the pulse bandwidth. The sampling rate must be an integer multiple of the PRF. Therefore, modify the sampling rate to satisfy the requirement.

pulse_bw = c/(2*range_res); % Pulse bandwidth pulse_width = 1/pulse_bw; % Pulse width fs = 2*pulse_bw; % Sampling rate n = ceil(fs/PRF); fs = n*PRF; waveform = phased.RectangularWaveform('PulseWidth',pulse_width,'PRF',PRF,... 'SampleRate',fs);

**Antennas and URA Array**

The array consists of short-dipole antenna elements. Use the `phased.ShortDipoleAntennaElement`

System object to create a short-dipole antenna oriented alont the *z*-axis.

antenna = phased.ShortDipoleAntennaElement(... 'FrequencyRange',[5e9,10e9],'AxisDirection','Z');

Define a 31-by-31 Taylor tapered uniform rectangular array using the `phased.URA`

System object. Set the size of the array using the number of rows, `numRows`

, and the number of columns, `numCols`

. The distance between elements, `d`

, is slightly smaller than one-half the wavelength, `lambda`

. Compute the array taper, `tw`

, using separate Taylor windows for the row and column directions. Obtain the Taylor weights using the `taylorwin`

function. Plot the 3-D array response using the array `pattern`

method.

numCols = 31; numRows = 31; lambda = c/fc; d = 0.9*lambda/2; % Nominal spacing wc = taylorwin(numCols); wr = taylorwin(numRows); tw = wr*wc'; array = phased.URA('Element',antenna,'Size',[numCols,numRows],... 'ElementSpacing',[d,d],'Taper',tw); pattern(array,fc,[-180:180],[-90:90],'CoordinateSystem','polar','Type','powerdb',... 'Polarization','V');

**Radar Platform Motion**

Next, set the position and motion of the radar platform in the `phased.Platform`

System object. The radar is assumed to be stationary and positioned at the origin `Velocity`

property to `[0,0,0]`

and the `InitialPosition`

property to `[0,0,0]`

. Set the `InitialOrientationAxes`

property to the identity matrix to align the radar platform coordinate axes with the global coordinate system.

radarPlatformAxes = [1 0 0;0 1 0;0 0 1]; radarplatform = phased.Platform('InitialPosition',[0;0;0],... 'Velocity',[0;0;0],'OrientationAxes',radarPlatformAxes);

**Transmitters and Receivers**

In radar, the signal propagates in the form of an electromagnetic wave. The signal is radiated and collected by the antennas used in the radar system. Associate the array with a radiator, `phased.Radiator`

, System object and two collector, `phased.Collector, System objects. Set the |WeightsInputPort`

property of the radiator to `true`

to enable dynamic steering of the transmitted signal at each execution of the radiator. Creating the two collectors allows for collection of both horizontal and vertical polarization components.

radiator = phased.Radiator('Sensor',array,'OperatingFrequency',fc,... 'PropagationSpeed',c,'CombineRadiatedSignals',true,... 'Polarization','Combined','WeightsInputPort',true); collector1 = phased.Collector('Sensor',array,'OperatingFrequency',fc,... 'PropagationSpeed',c,'Wavefront','Plane','Polarization','Combined',... 'WeightsInputPort',false); collector2 = phased.Collector('Sensor',array,'OperatingFrequency',fc,... 'PropagationSpeed',c,'Wavefront','Plane','Polarization','Combined',... 'WeightsInputPort',false);

Estimate the peak power needed in the `phased.Transmitter`

System object to calculate the desired radiated power levels. The transmitted peak power is the power required to achieve a minimum-detection SNR, `snr_min`

. You can determine the minimum SNR from the probability of detection,|pd|, and the probability of false alarm, `pfa`

, using the <function `albersheim`

function. Then, compute the peak power from the radar equation using the `radareqpow`

function. Among the inputs to this function are the overall signal gain, which is the sum of the transmitting element gain, `TransmitterGain`

and the array gain, `AG`

. Another input is the maximum detection range, `rangegate`

. Finally, you need to supply a target cross-section value, `tgt_rcs`

. A scalar radar cross section is used in this code section as an approximation even though the full polarization computation later uses a 2-by-2 radar cross section scattering matrix.

Estimate the total transmitted power to achieve a required detection SNR using all the pulses.

The SNR has contributions from the transmitting element gain as well as the array gain. Compute first an estimate of the array gain, then add the array gain to the transmitter gain to get the peak power which achieves the desired SNR.

Use an approximate target cross section of 1 for the radar equation even though the analysis calls for the full scattering matrix.

Set the maximum range to be equal to the value of 'rangegate' since targets outside that range are of no interest.

Compute the array gain as 10*log10(number of elements)

Assume each element has a gain of 20 dB.

snr_min = albersheim(pd, pfa, numpulses); AG = 10*log10(numCols*numRows); tgt_rcs = 1; TransmitterGain = 20; peak_power = radareqpow(lambda,rangegate,snr_min,waveform.PulseWidth,... 'RCS',tgt_rcs,'Gain',TransmitterGain + AG); transmitter = phased.Transmitter('PeakPower',peak_power,'Gain',TransmitterGain,... 'LossFactor',0,'InUseOutputPort',true,'CoherentOnTransmit',true);

**Define Target**

We want to simulate the pulse returns from a target that is rotating so that the scattering cross-section matrix changes from pulse to pulse. Create a rotating target object and a moving target platform. The rotating target is represented later as an angle-dependent scattering matrix. Rotation is in degrees/sec.

targetSpeed = 1000; targetVec = [-1;1;0]/sqrt(2); target = phased.RadarTarget('EnablePolarization',true,... 'Mode','Monostatic','ScatteringMatrixSource','Input port',... 'OperatingFrequency',fc); targetPlatformAxes = [1 0 0;0 1 0;0 0 1]; targetRotRate = 45; targetplatform = phased.Platform('InitialPosition',[3500.0; 0; 0],... 'Velocity', targetSpeed*targetVec);

**Other System objects**

Steering vector defined by

`phased.SteeringVector`

System object.Beamformer defined by

`phased.PhaseShiftBeamformer`

System object. The`DirectionSource`

property is set to`'Input Port'`

to enable the beamformer to always points towards the known target direction at each execution.Free-space propagator using the

`phased.FreeSpace`

System object.Receiver preamp model using the

`phased.ReceiverPreamp`

system object.

**Signal propagation**

Because the reflected signals are received by an array, use a beamformer pointing to the steering direction to obtain the combined signal.

steeringvector = phased.SteeringVector('SensorArray',array,'PropagationSpeed',c,... 'IncludeElementResponse',false); beamformer = phased.PhaseShiftBeamformer('SensorArray',array,... 'OperatingFrequency',fc,'PropagationSpeed',c,... 'DirectionSource','Input port'); channel = phased.FreeSpace('SampleRate',fs,... 'TwoWayPropagation',true,'OperatingFrequency',fc); % Define a receiver with receiver noise amplifier = phased.ReceiverPreamp('Gain',20,'LossFactor',0,'NoiseFigure',1,... 'ReferenceTemperature',290,'SampleRate',fs,'EnableInputPort',true,... 'PhaseNoiseInputPort',false,'SeedSource','Auto');

For such a large PRI and sampling rate, there will be too many samples per element. This will cause problems with the collector which has 961 channels. To keep the number of samples manageable, set a maximum range of 5 km. We know that the target is within this range.

This set of axes specifies the direction of the local coordinate axes with respect to the global coordinate system. This is the orientation of the target.

**Processing Loop**

Pre-allocate arrays for collecting data to be plotted.

sig_max_V = zeros(1,numpulses); sig_max_H = zeros(1,numpulses); tm_V = zeros(1,numpulses); tm_H = zeros(1,numpulses);

After all the System objects are created, loop over the number of pulses to create the reflected signals.

maxsamp = ceil(tmax*fs); fast_time_grid = [0:(maxsamp-1)]/fs; rotangle = 0.0; for m = 1:numpulses x = waveform(); % Generate pulse % Capture only samples within range gated x = x(1:maxsamp); [s, tx_status] = transmitter(x); % Create transmitted pulse % Move the radar platform and target platform. [radarPos,radarVel] = radarplatform(1/PRF); [targetPos,targetVel] = targetplatform(1/PRF); % Compute the known target angle [targetRng,targetAng] = rangeangle(targetPos,... radarPos,... radarPlatformAxes); % Compute the radar angle with respect to the target axes. [radarRng,radarAng] = rangeangle(radarPos,... targetPos,... targetPlatformAxes); % Calculate the steering vector designed to track the target sv = steeringvector(fc,targetAng); % Radiate the polarized signal toward the targat tsig1 = radiator(s,targetAng,radarPlatformAxes,conj(sv)); % Compute the two-way propagation loss (4*pi*R/lambda)^2 tsig2 = channel(tsig1,radarPos,targetPos,radarVel,targetVel); % Create a very simple model of a changing scattering matrix scatteringMatrix = [cosd(rotangle),0.5*sind(rotangle);... 0.5*sind(rotangle),cosd(rotangle)]; rsig1 = target(tsig2,radarAng,targetPlatformAxes,scatteringMatrix); % Reflect off target % Collect the vertical component of the radiation. rsig3V = collector1(rsig1,targetAng,radarPlatformAxes); % Collect the horizontal component of the radiation. This % second collector is rotated around the x-axis to be more % sensitive to horizontal polarization rsig3H = collector2(rsig1,targetAng,rotx(90)*radarPlatformAxes); % Add receiver noise to both sets of signals rsig4V = amplifier(rsig3V,~(tx_status>0)); % Receive signal rsig4H = amplifier(rsig3H,~(tx_status>0)); % Receive signal % Beamform the signal rsigV = beamformer(rsig4V,targetAng); % Beamforming rsigH = beamformer(rsig4H,targetAng); % Beamforming % Find the maximum returns for each pulse and store them in % a vector. Store the pulse received time as well. [sigmaxV,imaxV] = max(abs(rsigV)); [sigmaxH,imaxH] = max(abs(rsigH)); sig_max_V(m) = sigmaxV; sig_max_H(m) = sigmaxH; tm_V(m) = fast_time_grid(imaxV) + (m-1)*PRI; tm_H(m) = fast_time_grid(imaxH) + (m-1)*PRI; % Update the orientation of the target platform axes targetPlatformAxes = ... rotx(PRI*targetRotRate)*targetPlatformAxes; rotangle = rotangle + PRI*targetRotRate; end % Plot the vertical and horizontal polarization for each pulse as a % function of time. plot(tm_V,sig_max_V,'.') hold on plot(tm_H,sig_max_H,'r.') hold off xlabel('Time (sec)') ylabel('Amplitude') title('Vertical and Horizontal Polarization Components') legend('Vertical','Horizontal') grid on

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